zbMATH — the first resource for mathematics

New RBF collocation methods and kernel RBF with applications. (English) Zbl 1016.65094
Griebel, Michael (ed.) et al., Meshfree methods for partial differential equations. International workshop, Univ. Bonn, Germany, September 11-14, 2001. Berlin: Springer. Lect. Notes Comput. Sci. Eng. 26, 75-86 (2002).
Summary: A few novel Radial Basis Function (RBF) discretization schemes for partial differential equations are developed in this study. For boundary-type methods we derive the indirect and direct symmetric boundary knot methods. Based on the multiple reciprocity principle, the boundary particle method is introduced for general inhomogeneous problems without using inner nodes. For domain-type schemes, by using the Green integral we develop a novel Hermite RBF scheme called the modified Kansa method [cf. E. J. Konsa, Comput. Math. Appl. 19, No. 8/9, 147-161 (1990; Zbl 0850.76048)], which significantly reduces calculation errors at close-to-boundary nodes. To avoid Gibbs phenomenon, we present the least square RBF collocation scheme. Finally, five types of the kernel RBF are also briefly presented.
For the entire collection see [Zbl 0996.00042].

65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65N38 Boundary element methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations